Problem: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $t \neq 0$. $a = \dfrac{2t + 10}{4t^2 + 48t + 140} \times \dfrac{-3t^2 - 27t - 42}{t - 9} $
First factor out any common factors. $a = \dfrac{2(t + 5)}{4(t^2 + 12t + 35)} \times \dfrac{-3(t^2 + 9t + 14)}{t - 9} $ Then factor the quadratic expressions. $a = \dfrac {2(t + 5)} {4(t + 7)(t + 5)} \times \dfrac {-3(t + 7)(t + 2)} {t - 9} $ Then multiply the two numerators and multiply the two denominators. $a = \dfrac {2(t + 5) \times -3(t + 7)(t + 2) } { 4(t + 7)(t + 5) \times (t - 9)} $ $a = \dfrac {-6(t + 7)(t + 2)(t + 5)} {4(t + 7)(t + 5)(t - 9)} $ Notice that $(t + 7)$ and $(t + 5)$ appear in both the numerator and denominator so we can cancel them. $a = \dfrac {-6\cancel{(t + 7)}(t + 2)(t + 5)} {4\cancel{(t + 7)}(t + 5)(t - 9)} $ We are dividing by $t + 7$ , so $t + 7 \neq 0$ Therefore, $t \neq -7$ $a = \dfrac {-6\cancel{(t + 7)}(t + 2)\cancel{(t + 5)}} {4\cancel{(t + 7)}\cancel{(t + 5)}(t - 9)} $ We are dividing by $t + 5$ , so $t + 5 \neq 0$ Therefore, $t \neq -5$ $a = \dfrac {-6(t + 2)} {4(t - 9)} $ $ a = \dfrac{-3(t + 2)}{2(t - 9)}; t \neq -7; t \neq -5 $